Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Basic Probability Inequalities for Sums of Independent Random Variables
- 3 Uniform Convergence and Generalization Analysis
- 4 Empirical Covering Number Analysis and Symmetrization
- 5 Covering Number Estimates
- 6 Rademacher Complexity and Concentration Inequalities
- 7 Algorithmic Stability Analysis
- 8 Model Selection
- 9 Analysis of Kernel Methods
- 10 Additive and Sparse Models
- 11 Analysis of Neural Networks
- 12 Lower Bounds and Minimax Analysis
- 13 Probability Inequalities for Sequential Random Variables
- 14 Basic Concepts of Online Learning
- 15 Online Aggregation and Second-Order Algorithms
- 16 Multiarmed Bandits
- 17 Contextual Bandits
- 18 Reinforcement Learning
- Appendix A Basics of Convex Analysis
- Appendix B f-divergence of Probability Measures
- References
- Author Index
- Subject Index
12 - Lower Bounds and Minimax Analysis
Published online by Cambridge University Press: 20 July 2023
Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Basic Probability Inequalities for Sums of Independent Random Variables
- 3 Uniform Convergence and Generalization Analysis
- 4 Empirical Covering Number Analysis and Symmetrization
- 5 Covering Number Estimates
- 6 Rademacher Complexity and Concentration Inequalities
- 7 Algorithmic Stability Analysis
- 8 Model Selection
- 9 Analysis of Kernel Methods
- 10 Additive and Sparse Models
- 11 Analysis of Neural Networks
- 12 Lower Bounds and Minimax Analysis
- 13 Probability Inequalities for Sequential Random Variables
- 14 Basic Concepts of Online Learning
- 15 Online Aggregation and Second-Order Algorithms
- 16 Multiarmed Bandits
- 17 Contextual Bandits
- 18 Reinforcement Learning
- Appendix A Basics of Convex Analysis
- Appendix B f-divergence of Probability Measures
- References
- Author Index
- Subject Index
Summary
This chapter considers lower bounds for empirical processes and statistical estimation problems. We know that upper bounds for empirical processes and empirical risk minimization can be obtained from the covering number analysis. We show that, under suitable conditions, lower bounds can also be obtained using covering numbers.
- Type
- Chapter
- Information
- Mathematical Analysis of Machine Learning Algorithms , pp. 246 - 266Publisher: Cambridge University PressPrint publication year: 2023